Solve x^2-82x-4080=0

$${ x }^{ 2 } -82x-4080=0$$

Solve for x

$$x=\sqrt{5761}+41\approx 116.901251637$$

$$x=41-\sqrt{5761}\approx -34.901251637$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-82x-4080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -82 for b, and -4080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-82\right)±\sqrt{\left(-82\right)^{2}-4\left(-4080\right)}}{2}$$

Square -82.

$$x=\frac{-\left(-82\right)±\sqrt{6724-4\left(-4080\right)}}{2}$$

Multiply -4 times -4080.

$$x=\frac{-\left(-82\right)±\sqrt{6724+16320}}{2}$$

Add 6724 to 16320.

$$x=\frac{-\left(-82\right)±\sqrt{23044}}{2}$$

Take the square root of 23044.

$$x=\frac{-\left(-82\right)±2\sqrt{5761}}{2}$$

The opposite of -82 is 82.

$$x=\frac{82±2\sqrt{5761}}{2}$$

Now solve the equation x=\frac{82±2\sqrt{5761}}{2} when ± is plus. Add 82 to 2\sqrt{5761}.

$$x=\frac{2\sqrt{5761}+82}{2}$$

Divide 82+2\sqrt{5761} by 2.

$$x=\sqrt{5761}+41$$

Now solve the equation x=\frac{82±2\sqrt{5761}}{2} when ± is minus. Subtract 2\sqrt{5761} from 82.

$$x=\frac{82-2\sqrt{5761}}{2}$$

Divide 82-2\sqrt{5761} by 2.

$$x=41-\sqrt{5761}$$

The equation is now solved.

$$x=\sqrt{5761}+41 x=41-\sqrt{5761}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-82x-4080=0$$

Add 4080 to both sides of the equation.

$$x^{2}-82x-4080-\left(-4080\right)=-\left(-4080\right)$$

Subtracting -4080 from itself leaves 0.

$$x^{2}-82x=-\left(-4080\right)$$

Subtract -4080 from 0.

$$x^{2}-82x=4080$$

Divide -82, the coefficient of the x term, by 2 to get -41. Then add the square of -41 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-82x+\left(-41\right)^{2}=4080+\left(-41\right)^{2}$$

Square -41.

$$x^{2}-82x+1681=4080+1681$$

Add 4080 to 1681.

$$x^{2}-82x+1681=5761$$

Factor x^{2}-82x+1681. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-41\right)^{2}=5761$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-41\right)^{2}}=\sqrt{5761}$$

Simplify.

$$x-41=\sqrt{5761} x-41=-\sqrt{5761}$$

Add 41 to both sides of the equation.

$$x=\sqrt{5761}+41 x=41-\sqrt{5761}$$